Question

a curved wedge is cut from a cylinder of radius 3 by two planes. One plane is perpendicular to the axis of the cylinder. The second plane crosses the first plane at 45°angle at the center of the cylinder. Find the volume of the wedge.​

Answer 1

volume of wedeg is 18 sq unit .

see figure, here we have drawn a typical cross sectional area which is perpendicular to x-axis.

from figure it is clear that cross sectional area at x-axis is rectangle.

let height of rectangle is x

then, width of it will be 2√(3² - x²) [ from Pythagoras theorem ]

= 2√(9 - x²)

so, cross sectional area , dA = x(2x√(9 - x²)

as volume is given as $V=\int\limits^a_b{A(x)}\,dx$

here, A(x) = 2x√(9 - x²) , a = 3 and b = 0

so, V = $\int\limits^3_0{(2x\sqrt{9-x^2})}\,dx$

[using application, $\int{x\sqrt{a^2-x^2}}\,dx=-\frac{1}{3}(a^2-x^2)^{3/2}$ ]

= $-\frac{2}{3}\left[(9-x^2)^{3/2}\right]^3_0$

= -2/3 × [0 - (9)^(3/2)]

= -2/3 × -27

= 18

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